Open k-monopolies in graphs: complexity and related concepts

Closed monopolies in graphs have a quite long range of applications in several problems related to overcoming failures, since they frequently have some common approaches around the notion of majorities, for instance to consensus problems, diagnosis problems or voting systems. We introduce here open $k$-monopolies in graphs which are closely related to different parameters in graphs. Given a graph $G=(V,E)$ and $X\subseteq V$, if $\delta_X(v)$ is the number of neighbors $v$ has in $X$, $k$ is an integer and $t$ is a positive integer, then we establish in this article a connection between the following three concepts: - Given a nonempty set $M\subseteq V$ a vertex $v$ of $G$ is said to be $k$-controlled by $M$ if $\delta_M(v)\ge \frac{\delta_V(v)}{2}+k$. The set $M$ is called an open $k$-monopoly for $G$ if it $k$-controls every vertex $v$ of $G$. - A function $f: V\rightarrow \{-1,1\}$ is called a signed total $t$-dominating function for $G$ if $f(N(v))=\sum_{v\in N(v)}f(v)\geq t$ for all $v\in V$. - A nonempty set $S\subseteq V$ is a global (defensive and offensive) $k$-alliance in $G$ if $\delta_S(v)\ge \delta_{V-S}(v)+k$ holds for every $v\in V$. In this article we prove that the problem of computing the minimum cardinality of an open $0$-monopoly in a graph is NP-complete even restricted to bipartite or chordal graphs. In addition we present some general bounds for the minimum cardinality of open $k$-monopolies and we derive some exact values.Comment: 18 pages, Discrete Mathematics & Theoretical Computer Science (2016

Reflecting in and on post-observation feedback in initial teacher training on certificate courses

This article examines evidence from two studies that concern the nature of post-observation feedback in certificate courses for teaching English to speakers of other languages. It uncovers the main characteristics of these meetings and asks whether there is evidence of reflection in these contexts. In considering reasons why making space for reflection is potentially difficult, the paper also examines the relationship and the role of assessment criteria and how these may impact on opportunities for reflection. The final part of the paper considers how a more reflective approach could be promoted in feedback conferences

Women’s Rights in Kenya since Independence: The Complexities of Kenya’s Legal System and the Opportunities of Civic Engagement

Since Kenya gained independence from Britain in 1963, women’s rights in the country have made slow gains and suffered some setbacks. However, the rights of women and their guaranteed participation in politics was outlined in Kenya’s 2010 Constitution. This paper will survey some of those gains as well as describe the social backlash experienced by women leaders who have been trailblazers in post-colonial Kenyan politics

Responding to class theft: Theoretical and empirical links to critical management studies

Redrafted submission for inclusion in Remarx Section of Rethinking MarxismThis paper suggests closer linkages between the fields of Postmodern Class Analysis (PCA) and Critical Management Studies (CMS)2 are possible. It argues that CMS might contribute to the empirical engagement with the over-determined relations between class and non-class processes in work organizations (this appears to have received relatively little attention in PCA) and that PCA's theoretical and conceptual commitments may provide one means for CMS to engage in class analysis. CMS's focus on power and symbolic relations has led to the relative neglect of exploitation and class, in surplus terms. Both fields share similar although not identical political and ethical commitments

Centroidal bases in graphs

We introduce the notion of a centroidal locating set of a graph $G$, that is, a set $L$ of vertices such that all vertices in $G$ are uniquely determined by their relative distances to the vertices of $L$. A centroidal locating set of $G$ of minimum size is called a centroidal basis, and its size is the centroidal dimension $CD(G)$. This notion, which is related to previous concepts, gives a new way of identifying the vertices of a graph. The centroidal dimension of a graph $G$ is lower- and upper-bounded by the metric dimension and twice the location-domination number of $G$, respectively. The latter two parameters are standard and well-studied notions in the field of graph identification. We show that for any graph $G$ with $n$ vertices and maximum degree at least~2, $(1+o(1))\frac{\ln n}{\ln\ln n}\leq CD(G) \leq n-1$. We discuss the tightness of these bounds and in particular, we characterize the set of graphs reaching the upper bound. We then show that for graphs in which every pair of vertices is connected via a bounded number of paths, $CD(G)=\Omega\left(\sqrt{|E(G)|}\right)$, the bound being tight for paths and cycles. We finally investigate the computational complexity of determining $CD(G)$ for an input graph $G$, showing that the problem is hard and cannot even be approximated efficiently up to a factor of $o(\log n)$. We also give an $O\left(\sqrt{n\ln n}\right)$-approximation algorithm